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<title>Atlas software user guide -- Real forms</title>
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<h2>Real forms</h2>
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<i>Last updated: November 26, 2005</i>
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Let G be a connected complex reductive algebraic group. According to general
principles in algebraic geometry, a <i>real form</i> of G is an antilinear
involution of G; the set of fixed points of the involution is denoted 
G(<b>R</b>) and
is a real reductive group, often non-connected. It turns out that there is
a bijective correspondence between G-conjugacy classes of antilinear 
involutions of G, and of ordinary involutions of G as a complex algebraic
group; to set this up, one chooses a compact real form of G, and conjugates
both types of involutions to ones that commute with the chosen compact
antilinear involution.
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<i>Therefore, in this program we always represent real
forms through ordinary involutions of G.</i>
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To relate our data to perhaps more familiar objects, one should think of our
involution &theta; as the complexification of a Cartan involution for 
G(<b>R</b>). In particular, the compact form of G corresponds to taking 
&theta; = Id, the identity involution. Similarly, one should think of the group
K of &theta;-fixed points in G as the complexification of a maximal compact 
subgroup of G(<b>R</b>).
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We consider two real forms to be equivalent if they are conjugate under G
(a word of caution: the natural notion of isomorphism of real algebraic groups 
would translate to considering real forms up to conjugacy in the full
automorphism group of G; however it is conjugacy under G that is the 
appropriate one for our purposes. An example where the difference is apparent
is the case of <a href="Deven.html">type D_2m</a>; a more elementary example
is G = <b>SL</b>(2).<b>SL</b>(2), where <b>SU</b>(2).<b>SL</b>(2,<b>R</b>) and 
<b>SL</b>(2,<b>R</b>).<b>SU</b>(2) are obviously isomorphic groups, but not
equivalent real forms in our sense.
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In fact, what we have called real forms so far should be called <i>weak</i>
real forms. There is a more subtle notion of <a href="strongreal.html">
strong</a> real form, which should be thought of as some kind of 
&#8220;lifting&#8221; of the notion of a weak real form (indeed, when G is
adjoint, the two notions coincide.) It turns out that the classifications of 
strong and weak real forms can be readily computed from the internal data that 
we keep in the program; in substance, the classifications amount to the 
computation of the orbits of an action of a subgroup of the Weyl group W of G 
on a small finite set (the precise statement about the classification of strong
real forms may be found <a href="strongreal.html">here</a>.) The classification
of weak real forms is printed out by the 
&#8220;showrealforms&#8221; command, and is used consistently in all instances
where a choice of real form must be made. The various combinatorial types
of strong real form packets are output by the &#8220;strongreal&#8221;
command.
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